References¶
Related work¶
Citation keys refer to docs/references.bib. Every
entry with a DOI was checked against Crossref and its BibTeX taken from
Crossref by docs/verify_refs.py (119 entries). The 39 entries without a
Crossref DOI (conference papers, arXiv, software, reports) are written by
hand in docs/refs.json; the same script confirms each one from an
independent source: the arXiv record, the DOI record, OpenAlex, the GitHub
repository, or the entry's title on the publisher's page. The CEC 2006
report has no online record and was checked against its title page.
tests/test_citations.py fails on any [key] in the repo's .py, .md or
notebook sources that is not in the bibliography.
McCormick relaxations of code. The relaxation arithmetic is McCormick's
(McCormick 1976), in the generalized and subgradient form of
(Mitsos et al. 2009; Scott et al. 2011). That work relaxes whole programs
by operator overloading (MC++ [chachuatmcpp, chachuat2015settheoretic]); MAiNGO
(Bongartz et al. 2018; Bongartz & Mitsos 2017) and EAGO.jl
(Wilhelm & Stuber 2020) are the solvers built on it. Reduced-space branching on
the original variables follows (Epperly & Pistikopoulos 1997). Convergence theory:
(Bompadre & Mitsos 2011; Najman & Mitsos 2016). Possible tightenings:
multivariate rules (Tsoukalas & Mitsos 2014), subgradient bound tightening
(Najman & Mitsos 2019), reverse propagation (Wechsung et al. 2015), LP from
linearizations (Najman et al. 2021), Taylor and McCormick–Taylor models
(Makino & Berz 2003; Sahlodin & Chachuat 2011; Bompadre et al. 2012), DAG interval propagation (Schichl & Neumaier 2005),
differentiable relaxations (Khan et al. 2016), and relaxations of implicit
functions (Stuber et al. 2014), a route to relaxing lax.while_loop.
Embedded ML models: (Schweidtmann & Mitsos 2018; Schweidtmann et al. 2021; Wilhelm et al. 2022). jaxglobal's difference: no
operator overloading or code generation. The relaxation is a re-interpretation
of the jaxpr, so jit, vmap and grad apply to the bounding itself.
αBB and interval Hessians. αBB (Androulakis et al. 1995; Adjiman & Floudas 1996; Adjiman et al. 1998; Adjiman et al. 1998) with tighter α
(Hladík 2014; Nerantzis & Adjiman 2019; Kazazakis & Adjiman 2018; Skjäl et al. 2012) and univariate underestimators
(Gounaris & Floudas 2008; Meyer & Floudas 2005). The closest prior art to evaluating
grad(grad(primitive)) on intervals is AD-based interval Hessian bounding
(Mönnigmann 2008; Schulze Darup et al. 2013; Schulze Darup & Mönnigmann 2016; Hladík 2017). We found no prior work that applies αBB automatically per
primitive to every nonconvex primitive of a traced program.
GPU and parallel global optimization. The most direct competitors: Gottlieb, Xu & Stuber generate McCormick kernels from Julia/Symbolics for GPU-batched lower bounding (Gottlieb et al. 2024; Gottlieb & Stuber 2026); Zhang et al. add GPU interval bounding to MAiNGO (Zhang et al. 2025). Earlier GPU/parallel B&B: (Barkalov & Gergel 2016; Bagóczki & Bánhelyi 2017; Gillner & Auer 2024; Gendron & Crainic 1994; Crainic et al. 2006; Gmys et al. 2016). Spatial B&B and its convergence theory: (Falk & Soland 1969; Horst & Tuy 1990; Smith & Pantelides 1999; Kannan & Barton 2017). Rigorous and interval global optimization: (Neumaier 2004; Kearfott 1996; Global Optimization Using Interval Analysis 2003). Branching rules: (Achterberg et al. 2005).
Bound propagation in ML and jaxpr interpreters. jaxpr-interpreting bounders already exist, but for other jobs: - jax_verify (Stanforth et al. 2020): NN verification, with bilinear McCormick and B&B over activation/input splits. - AutoBound (Streeter & Dillon 2022; Streeter & Dillon 2023): Taylor enclosures, with a toy B&B. - immrax (Harapanahalli et al. 2024): interval reachability.
The verification literature has GPU-batched bounding in B&B: - linear bounds: IBP (Gowal et al. 2018), CROWN (Zhang et al. 2018), DeepPoly (Singh et al. 2019), auto_LiRPA (Xu et al. 2020); - B&B: α/β-CROWN (Xu et al. 2021; Wang et al. 2021), GenBaB (Shi et al. 2025), (Bunel et al. 2020); - dual and SDP relaxations: (Dvijotham et al. 2018; Dvijotham et al. 2020; Bunel et al. 2020); - relaxation-gap results: (Salman et al. 2019; Tjandraatmadja et al. 2020).
AD at kinks can return values that are not subgradients (Kakade & Lee 2018; Lee et al. 2020; Bolte & Pauwels 2020; Bolte & Pauwels 2020). This is the failure mode behind jaxglobal's tie-aware rules (see Lessons). Nonsmooth implicit differentiation: (Bolte et al. 2024). Principled fixes: abs-normal forms (Griewank 2013; Griewank et al. 2015) and lexicographic derivatives (Nesterov 2005; Khan & Barton 2013; Khan & Barton 2015).
LP bounds. Node bounds come from cuts (Kelley 1960; Frank & Wolfe 1956; Jaggi 2013), fed to an interior-point LP (Mehrotra 1992; Wright 1997). The bound is made safe by weak duality with box bounds, following (Neumaier & Shcherbina 2004; Jansson 2004; Keil & Jansson 2006). That safety is what allows first-order GPU LP solvers in the tree: PDLP (Applegate et al. 2021; Applegate et al. 2026; Applegate et al. 2022; Applegate et al. 2024), cuPDLP (Lu & Yang 2025; Lu & Yang 2025), and MPAX in JAX (Lu et al. 2024). Batched differentiable solvers: (Amos & Kolter 2017; Agrawal et al. 2019).
Global solvers and bound tightening (comparison baselines and roadmap
targets):
- solvers: BARON (Tawarmalani & Sahinidis 2005; Khajavirad & Sahinidis 2018), SCIP
(Vigerske & Gleixner 2017; Bestuzheva et al. 2023; Bolusani et al. 2024; Hojny et al. 2025),
Couenne (Belotti et al. 2009), ANTIGONE (Misener & Floudas 2014),
LindoGlobal (Lin & Schrage 2009);
- bound tightening: FBBT (Belotti et al. 2010), domain reduction
(Puranik & Sahinidis 2017; Ryoo & Sahinidis 1996);
- checking solver claims against reference values: (Neumaier et al. 2005),
which benchmarks/report.py follows.
Local solvers and infrastructure. - local solvers: pounce derives from Ipopt (Wächter & Biegler 2005); L-BFGS(-B) (Liu & Nocedal 1989; Byrd et al. 1995), SLSQP (Kraft 1988); - augmented Lagrangian: (Hestenes 1969; Powell 1969); - tools: SciPy (Virtanen et al. 2020), JAX (Bradbury et al. 2018; Frostig et al. 2018), optax (DeepMind et al. 2020), AD (Griewank & Walther 2008); - modeling for the comparisons: Pyomo (Hart et al. 2011; Bynum et al. 2021), GAMS (Brook et al. 1988); - related MINLP work by the author: discopt (Kitchin 2026), which uses an algebraic modeling API rather than tracing; - ML-embedded optimization: (Ceccon et al. 2022; Tsay et al. 2021; Kronqvist et al. 2025; Anderson et al. 2020; Fischetti & Jo 2018; Grimstad & Andersson 2019).
Test problems. - sources: (Floudas et al. 1999; Ryoo & Sahinidis 1995; Haverly 1978; Michalewicz & Schoenauer 1996; Liang et al. 2006; Surjanovic & Bingham 2013; Jamil & Yang 2013); - MINLPLib (Bussieck et al. 2003).
Citing and license¶
jaxglobal is released under the MIT license (LICENSE). To cite
it, use the metadata in CITATION.cff (GitHub's "Cite this
repository" button). Each GitHub release is archived on Zenodo. The DOI
10.5281/zenodo.23478503 always
resolves to the latest release; version 0.1.0 is
10.5281/zenodo.23478504.
@software{kitchin2026jaxglobal,
author = {Kitchin, John R.},
title = {jaxglobal: deterministic global optimization of plain {JAX} functions},
year = {2026},
publisher = {Zenodo},
version = {0.1.0},
doi = {10.5281/zenodo.23478503},
url = {https://github.com/jkitchin/jax-go}
}
Bibliography¶
Generated from docs/references.bib by docs/readme_bib.py.
Achterberg, T., Koch, T., Martin, A. (2005). Branching rules revisited. Operations Research Letters 33(1), 42–54. doi:10.1016/j.orl.2004.04.002
Adjiman, C. S., Floudas, C. A. (1996). Rigorous convex underestimators for general twice-differentiable problems. Journal of Global Optimization 9(1), 23–40. doi:10.1007/bf00121749
Adjiman, C., Dallwig, S., Floudas, C., Neumaier, A. (1998). A global optimization method, αBB, for general twice-differentiable constrained NLPs — I. Theoretical advances. Computers & Chemical Engineering 22(9), 1137–1158. doi:10.1016/s0098-1354(98)00027-1
Adjiman, C., Androulakis, I., Floudas, C. (1998). A global optimization method, αBB, for general twice-differentiable constrained NLPs—II. Implementation and computational results. Computers & Chemical Engineering 22(9), 1159–1179. doi:10.1016/s0098-1354(98)00218-x
Agrawal, A., Amos, B., Barratt, S., Boyd, S., Diamond, S., Kolter, J. Z. (2019). Differentiable Convex Optimization Layers. Advances in Neural Information Processing Systems 32. https://proceedings.neurips.cc/paper/2019/hash/9ce3c52fc54362e22053399d3181c638-Abstract.html
Amos, B., Kolter, J. Z. (2017). OptNet: Differentiable Optimization as a Layer in Neural Networks. Proceedings of the 34th International Conference on Machine Learning 70, 136–145. https://proceedings.mlr.press/v70/amos17a.html
Anderson, R., Huchette, J., Ma, W., Tjandraatmadja, C., Vielma, J. P. (2020). Strong mixed-integer programming formulations for trained neural networks. Mathematical Programming 183(1-2), 3–39. doi:10.1007/s10107-020-01474-5
Androulakis, I. P., Maranas, C. D., Floudas, C. A. (1995). αBB: A global optimization method for general constrained nonconvex problems. Journal of Global Optimization 7(4), 337–363. doi:10.1007/bf01099647
Applegate, D., Díaz, M., Hinder, O., Lu, H., Lubin, M., O'Donoghue, B., Schudy, W. (2021). Practical Large-Scale Linear Programming using Primal-Dual Hybrid Gradient. Advances in Neural Information Processing Systems 34. https://proceedings.neurips.cc/paper/2021/hash/a8fbbd3b11424ce032ba813493d95ad7-Abstract.html
Applegate, D., Hinder, O., Lu, H., Lubin, M. (2022). Faster first-order primal-dual methods for linear programming using restarts and sharpness. Mathematical Programming 201(1-2), 133–184. doi:10.1007/s10107-022-01901-9
Applegate, D., Díaz, M., Lu, H., Lubin, M. (2024). Infeasibility Detection with Primal-Dual Hybrid Gradient for Large-Scale Linear Programming. SIAM Journal on Optimization 34(1), 459–484. doi:10.1137/22m1510467
Applegate, D., Díaz, M., Hinder, O., Lu, H., Lubin, M., O’Donoghue, B., Schudy, W. (2026). PDLP: a practical first-order method for large-scale linear programming. Mathematical Programming Computation. doi:10.1007/s12532-026-00309-2
Bagóczki, Z., Bánhelyi, B. (2017). A Parallel Interval Arithmetic-based Reliable Computing Method on a GPU. Acta Cybernetica 23(2), 491–501. doi:10.14232/actacyb.23.2.2017.4
Barkalov, K., Gergel, V. (2016). Parallel global optimization on GPU. Journal of Global Optimization 66(1), 3–20. doi:10.1007/s10898-016-0411-y
Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A. (2009). Branching and bounds tightening techniques for non-convex MINLP. Optimization Methods and Software 24(4-5), 597–634. doi:10.1080/10556780903087124
Belotti, P., Cafieri, S., Lee, J., Liberti, L. (2010). Feasibility-Based Bounds Tightening via Fixed Points. Combinatorial Optimization and Applications, 65–76. doi:10.1007/978-3-642-17458-2_7
Bestuzheva, K., Chmiela, A., Müller, B., Serrano, F., Vigerske, S., Wegscheider, F. (2023). Global optimization of mixed-integer nonlinear programs with SCIP 8. Journal of Global Optimization 91(2), 287–310. doi:10.1007/s10898-023-01345-1
Bolte, J., Pauwels, E. (2020). A mathematical model for automatic differentiation in machine learning. Advances in Neural Information Processing Systems 33.
Bolte, J., Pauwels, E. (2020). Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning. Mathematical Programming 188(1), 19–51. doi:10.1007/s10107-020-01501-5
Bolte, J., Pauwels, E., Silveti-Falls, A. (2024). Differentiating Nonsmooth Solutions to Parametric Monotone Inclusion Problems. SIAM Journal on Optimization 34(1), 71–97. doi:10.1137/22m1541630
Bolusani, S., Besançon, M., Bestuzheva, K., Chmiela, A., Dionísio, J., et al. (2024). The SCIP Optimization Suite 9.0. https://arxiv.org/abs/2402.17702
Bompadre, A., Mitsos, A. (2011). Convergence rate of McCormick relaxations. Journal of Global Optimization 52(1), 1–28. doi:10.1007/s10898-011-9685-2
Bompadre, A., Mitsos, A., Chachuat, B. (2012). Convergence analysis of Taylor models and McCormick-Taylor models. Journal of Global Optimization 57(1), 75–114. doi:10.1007/s10898-012-9998-9
Bongartz, D., Mitsos, A. (2017). Deterministic global optimization of process flowsheets in a reduced space using McCormick relaxations. Journal of Global Optimization 69(4), 761–796. doi:10.1007/s10898-017-0547-4
Bongartz, D., Najman, J., Sass, S., Mitsos, A. (2018). MAiNGO – McCormick-based Algorithm for mixed-integer Nonlinear Global Optimization. http://permalink.avt.rwth-aachen.de/?id=729717
Bradbury, J., Frostig, R., Hawkins, P., Johnson, M. J., Leary, C., Maclaurin, D., Necula, G., Paszke, A., et al. (2018). JAX: composable transformations of Python+NumPy programs. Software, http://github.com/jax-ml/jax.
Brook, A., Kendrick, D., Meeraus, A. (1988). GAMS, a user’s guide. ACM SIGNUM Newsletter 23(3-4), 10–11. doi:10.1145/58859.58863
Bunel, R., Lu, J., Turkaslan, I., Torr, P. H. S., Kohli, P., Kumar, M. P. (2020). Branch and Bound for Piecewise Linear Neural Network Verification. Journal of Machine Learning Research 21(42), 1–39.
Bunel, R., De Palma, A., Desmaison, A., Dvijotham, K., Kohli, P., Torr, P. H. S., Kumar, M. P. (2020). Lagrangian Decomposition for Neural Network Verification. Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI) 124, 370–379.
Bussieck, M. R., Drud, A. S., Meeraus, A. (2003). MINLPLib—A Collection of Test Models for Mixed-Integer Nonlinear Programming. INFORMS Journal on Computing 15(1), 114–119. doi:10.1287/ijoc.15.1.114.15159
Bynum, M. L., Hackebeil, G. A., Hart, W. E., Laird, C. D., Nicholson, B. L., Siirola, J. D., Watson, J., Woodruff, D. L. (2021). Pyomo — Optimization Modeling in Python. Springer Optimization and Its Applications. doi:10.1007/978-3-030-68928-5
Byrd, R. H., Lu, P., Nocedal, J., Zhu, C. (1995). A Limited Memory Algorithm for Bound Constrained Optimization. SIAM Journal on Scientific Computing 16(5), 1190–1208. doi:10.1137/0916069
Ceccon, F., Jalving, J., Haddad, J., Thebelt, A., Tsay, C., Laird, C. D., Misener, R. (2022). OMLT: Optimization & Machine Learning Toolkit. Journal of Machine Learning Research 23(349), 1–8. https://jmlr.org/papers/v23/22-0277.html
Crainic, T. G., Le Cun, B., Roucairol, C. (2006). Parallel Branch‐and‐Bound Algorithms. Parallel Combinatorial Optimization, 1–28. doi:10.1002/9780470053928.ch1
DeepMind, Babuschkin, I., et al. (2020). The DeepMind JAX Ecosystem. https://github.com/google-deepmind.
Dvijotham, K., Stanforth, R., Gowal, S., Mann, T., Kohli, P. (2018). A Dual Approach to Scalable Verification of Deep Networks. Proceedings of the Thirty-Fourth Conference on Uncertainty in Artificial Intelligence (UAI), 550–559.
Dvijotham, K., Stanforth, R., Gowal, S., Qin, C., De, S., Kohli, P. (2020). Efficient Neural Network Verification with Exactness Characterization. Proceedings of the 35th Uncertainty in Artificial Intelligence Conference 115, 497–507. https://proceedings.mlr.press/v115/dvijotham20a.html
Epperly, T. G. W., Pistikopoulos, E. N. (1997). A Reduced Space Branch and Bound Algorithm for Global optimization. Journal of Global Optimization 11(3), 287–311. doi:10.1023/a:1008212418949
Falk, J. E., Soland, R. M. (1969). An Algorithm for Separable Nonconvex Programming Problems. Management Science 15(9), 550–569. doi:10.1287/mnsc.15.9.550
Fischetti, M., Jo, J. (2018). Deep neural networks and mixed integer linear optimization. Constraints 23(3), 296–309. doi:10.1007/s10601-018-9285-6
Floudas, C. A., Pardalos, P. M., Adjiman, C. S., Esposito, W. R., Gümüş, Z. H., Harding, S. T., Klepeis, J. L., Meyer, C. A., et al. (1999). Handbook of Test Problems in Local and Global Optimization. Nonconvex Optimization and Its Applications. doi:10.1007/978-1-4757-3040-1
Frank, M., Wolfe, P. (1956). An algorithm for quadratic programming. Naval Research Logistics Quarterly 3(1-2), 95–110. doi:10.1002/nav.3800030109
Frostig, R., Johnson, M. J., Leary, C. (2018). Compiling machine learning programs via high-level tracing. SysML Conference. https://research.google/pubs/compiling-machine-learning-programs-via-high-level-tracing/
Gendron, B., Crainic, T. G. (1994). Parallel Branch-and-Branch Algorithms: Survey and Synthesis. Operations Research 42(6), 1042–1066. doi:10.1287/opre.42.6.1042
Gillner, L., Auer, E. (2024). GPU-Accelerated, Interval-Based Parameter Identification Methods Illustrated Using the Two-Compartment Problem. Acta Cybernetica 26(4), 913–932. doi:10.14232/actacyb.306774
(2003). Global Optimization Using Interval Analysis. CRC Press. doi:10.1201/9780203026922
Gmys, J., Mezmaz, M., Melab, N., Tuyttens, D. (2016). A GPU-based Branch-and-Bound algorithm using Integer–Vector–Matrix data structure. Parallel Computing 59, 119–139. doi:10.1016/j.parco.2016.01.008
Gottlieb, R. X., Stuber, M. D. (2026). Automatic generation of GPU kernels for evaluators of McCormick-based relaxations and subgradients. Mathematical Programming Computation. doi:10.1007/s12532-026-00339-w
Gottlieb, R. X., Xu, P., Stuber, M. D. (2024). Automatic source code generation for deterministic global optimization with parallel architectures. Optimization Methods and Software 41(2), 308–346. doi:10.1080/10556788.2024.2396297
Gounaris, C. E., Floudas, C. A. (2008). Tight convex underestimators for \(C^2\)-continuous problems: I. univariate functions. Journal of Global Optimization 42(1), 51–67. doi:10.1007/s10898-008-9287-9
Gowal, S., Dvijotham, K., Stanforth, R., Bunel, R., Qin, C., Uesato, J., Arandjelovic, R., Mann, T., et al. (2018). On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models. https://arxiv.org/abs/1810.12715
Griewank, A., Walther, A. (2008). Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Second Edition. Society for Industrial and Applied Mathematics. doi:10.1137/1.9780898717761
Griewank, A. (2013). On stable piecewise linearization and generalized algorithmic differentiation. Optimization Methods and Software 28(6), 1139–1178. doi:10.1080/10556788.2013.796683
Griewank, A., Bernt, J., Radons, M., Streubel, T. (2015). Solving piecewise linear systems in abs-normal form. Linear Algebra and its Applications 471, 500–530. doi:10.1016/j.laa.2014.12.017
Grimstad, B., Andersson, H. (2019). ReLU networks as surrogate models in mixed-integer linear programs. Computers & Chemical Engineering 131, 106580. doi:10.1016/j.compchemeng.2019.106580
Harapanahalli, A., Jafarpour, S., Coogan, S. (2024). immrax: A Parallelizable and Differentiable Toolbox for Interval Analysis and Mixed Monotone Reachability in JAX. IFAC-PapersOnLine 58(11), 75–80. doi:10.1016/j.ifacol.2024.07.428
Hart, W. E., Watson, J., Woodruff, D. L. (2011). Pyomo: modeling and solving mathematical programs in Python. Mathematical Programming Computation 3(3), 219–260. doi:10.1007/s12532-011-0026-8
Haverly, C. A. (1978). Studies of the behavior of recursion for the pooling problem. ACM SIGMAP Bulletin, 19–28. doi:10.1145/1111237.1111238
Hestenes, M. R. (1969). Multiplier and gradient methods. Journal of Optimization Theory and Applications 4(5), 303–320. doi:10.1007/bf00927673
Hladík, M. (2014). On the efficient Gerschgorin inclusion usage in the global optimization αBB method. Journal of Global Optimization 61(2), 235–253. doi:10.1007/s10898-014-0161-7
Hladík, M. (2017). The Effect of Hessian Evaluations in the Global Optimization αBB Method. Modeling, Simulation and Optimization of Complex Processes HPSC 2015, 67–79. doi:10.1007/978-3-319-67168-0_6
Hojny, C., Besançon, M., Bestuzheva, K., Borst, S., Dionísio, J., et al. (2025). The SCIP Optimization Suite 10.0. https://arxiv.org/abs/2511.18580
Horst, R., Tuy, H. (1990). Global Optimization. Springer Berlin Heidelberg. doi:10.1007/978-3-662-02598-7
Jaggi, M. (2013). Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization. Proceedings of the 30th International Conference on Machine Learning 28(1), 427–435. https://proceedings.mlr.press/v28/jaggi13.html
Jamil, M., Yang, X. S. (2013). A literature survey of benchmark functions for global optimisation problems. International Journal of Mathematical Modelling and Numerical Optimisation 4(2), 150. doi:10.1504/ijmmno.2013.055204
Jansson, C. (2004). Rigorous Lower and Upper Bounds in Linear Programming. SIAM Journal on Optimization 14(3), 914–935. doi:10.1137/s1052623402416839
Kakade, S. M., Lee, J. D. (2018). Provably correct automatic sub-differentiation for qualified programs. Advances in Neural Information Processing Systems 31. https://proceedings.neurips.cc/paper_files/paper/2018/hash/142c65e00f4f7cf2e6c4c996e34005df-Abstract.html
Kannan, R., Barton, P. I. (2017). Convergence-order analysis of branch-and-bound algorithms for constrained problems. Journal of Global Optimization 71(4), 753–813. doi:10.1007/s10898-017-0532-y
Kazazakis, N., Adjiman, C. S. (2018). Arbitrarily tight αBB underestimators of general non-linear functions over sub-optimal domains. Journal of Global Optimization 71(4), 815–844. doi:10.1007/s10898-018-0632-3
Kearfott, R. B. (1996). Rigorous Global Search: Continuous Problems. Nonconvex Optimization and Its Applications. doi:10.1007/978-1-4757-2495-0
Keil, C., Jansson, C. (2006). Computational Experience with Rigorous Error Bounds for the Netlib Linear Programming Library. Reliable Computing 12(4), 303–321. doi:10.1007/s11155-006-9004-7
Kelley, J. J. E. (1960). The Cutting-Plane Method for Solving Convex Programs. Journal of the Society for Industrial and Applied Mathematics 8(4), 703–712. doi:10.1137/0108053
Khajavirad, A., Sahinidis, N. V. (2018). A hybrid LP/NLP paradigm for global optimization relaxations. Mathematical Programming Computation 10(3), 383–421. doi:10.1007/s12532-018-0138-5
Khan, K. A., Barton, P. I. (2013). Evaluating an element of the Clarke generalized Jacobian of a composite piecewise differentiable function. ACM Transactions on Mathematical Software 39(4), 1–28. doi:10.1145/2491491.2491493
Khan, K. A., Barton, P. I. (2015). A vector forward mode of automatic differentiation for generalized derivative evaluation. Optimization Methods and Software 30(6), 1185–1212. doi:10.1080/10556788.2015.1025400
Khan, K. A., Watson, H. A. J., Barton, P. I. (2016). Differentiable McCormick relaxations. Journal of Global Optimization 67(4), 687–729. doi:10.1007/s10898-016-0440-6
Kitchin, J. R. (2026). discopt: a hybrid MINLP solver combining Rust, JAX, and Python. Zenodo. doi:10.5281/zenodo.19762815
Kraft, D. (1988). A software package for sequential quadratic programming.
Kronqvist, J., Misener, R., Tsay, C. (2025). P-split formulations: a class of intermediate formulations between big-M and convex hull for disjunctive constraints. Mathematical Programming 218(1-2), 57–94. doi:10.1007/s10107-025-02232-1
Lee, W., Yu, H., Rival, X., Yang, H. (2020). On correctness of automatic differentiation for non-differentiable functions. Advances in Neural Information Processing Systems 33.
Liang, J. J., Runarsson, T. P., Mezura-Montes, E., Clerc, M., Suganthan, P. N., Coello Coello, C. A., Deb, K. (2006). Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization. https://github.com/P-N-Suganthan/CEC2006/blob/master/technical_report.pdf
Lin, Y., Schrage, L. (2009). The global solver in the LINDO API. Optimization Methods and Software 24(4-5), 657–668. doi:10.1080/10556780902753221
Liu, D. C., Nocedal, J. (1989). On the limited memory BFGS method for large scale optimization. Mathematical Programming 45(1-3), 503–528. doi:10.1007/bf01589116
Lu, H., Yang, J. (2025). cuPDLP.jl: A GPU Implementation of Restarted Primal-Dual Hybrid Gradient for Linear Programming in Julia. Operations Research 73(6), 3440–3452. doi:10.1287/opre.2024.1069
Lu, H., Yang, J. (2025). A Practical and Optimal First-Order Method for Large-Scale Convex Quadratic Programming. Mathematical Programming 215(1-2), 771–808. doi:10.1007/s10107-025-02241-0
Lu, H., Peng, Z., Yang, J. (2024). MPAX: Mathematical Programming in JAX. doi:10.48550/arXiv.2412.09734
Makino, K., Berz, M. (2003). Taylor models and other validated functional inclusion methods. International Journal of Pure and Applied Mathematics 4(4), 379–456.
McCormick, G. P. (1976). Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems. Mathematical Programming 10(1), 147–175. doi:10.1007/bf01580665
Mehrotra, S. (1992). On the Implementation of a Primal-Dual Interior Point Method. SIAM Journal on Optimization 2(4), 575–601. doi:10.1137/0802028
Meyer, C. A., Floudas, C. A. (2005). Convex Underestimation of Twice Continuously Differentiable Functions by Piecewise Quadratic Perturbation: Spline αBB Underestimators. Journal of Global Optimization 32(2), 221–258. doi:10.1007/s10898-004-2704-9
Michalewicz, Z., Schoenauer, M. (1996). Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation 4(1), 1–32. doi:10.1162/evco.1996.4.1.1
Misener, R., Floudas, C. A. (2014). ANTIGONE: Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Equations. Journal of Global Optimization 59(2-3), 503–526. doi:10.1007/s10898-014-0166-2
Mitsos, A., Chachuat, B., Barton, P. I. (2009). McCormick-Based Relaxations of Algorithms. SIAM Journal on Optimization 20(2), 573–601. doi:10.1137/080717341
Mönnigmann, M. (2008). Efficient Calculation of Bounds on Spectra of Hessian Matrices. SIAM Journal on Scientific Computing 30(5), 2340–2357. doi:10.1137/070704186
Najman, J., Mitsos, A. (2016). Convergence analysis of multivariate McCormick relaxations. Journal of Global Optimization 66(4), 597–628. doi:10.1007/s10898-016-0408-6
Najman, J., Mitsos, A. (2019). Tighter McCormick relaxations through subgradient propagation. Journal of Global Optimization 75(3), 565–593. doi:10.1007/s10898-019-00791-0
Najman, J., Bongartz, D., Mitsos, A. (2021). Linearization of McCormick relaxations and hybridization with the auxiliary variable method. Journal of Global Optimization 80(4), 731–756. doi:10.1007/s10898-020-00977-x
Nerantzis, D., Adjiman, C. S. (2019). Tighter αBB relaxations through a refinement scheme for the scaled Gerschgorin theorem. Journal of Global Optimization 73(3), 467–483. doi:10.1007/s10898-018-0718-y
Nesterov, Y. (2005). Lexicographic differentiation of nonsmooth functions. Mathematical Programming 104(2-3), 669–700. doi:10.1007/s10107-005-0633-0
Neumaier, A., Shcherbina, O. (2004). Safe bounds in linear and mixed-integer linear programming. Mathematical Programming 99(2), 283–296. doi:10.1007/s10107-003-0433-3
Neumaier, A. (2004). Complete search in continuous global optimization and constraint satisfaction. Acta Numerica 13, 271–369. doi:10.1017/s0962492904000194
Neumaier, A., Shcherbina, O., Huyer, W., Vinkó, T. (2005). A comparison of complete global optimization solvers. Mathematical Programming 103(2), 335–356. doi:10.1007/s10107-005-0585-4
Powell, M. J. D. (1969). A method for nonlinear constraints in minimization problems. Optimization, 283–298.
Puranik, Y., Sahinidis, N. V. (2017). Domain reduction techniques for global NLP and MINLP optimization. Constraints 22(3), 338–376. doi:10.1007/s10601-016-9267-5
Ryoo, H., Sahinidis, N. (1995). Global optimization of nonconvex NLPs and MINLPs with applications in process design. Computers & Chemical Engineering 19(5), 551–566. doi:10.1016/0098-1354(94)00097-2
Ryoo, H. S., Sahinidis, N. V. (1996). A branch-and-reduce approach to global optimization. Journal of Global Optimization 8(2), 107–138. doi:10.1007/bf00138689
Sahlodin, A., Chachuat, B. (2011). Convex/concave relaxations of parametric ODEs using Taylor models. Computers & Chemical Engineering 35(5), 844–857. doi:10.1016/j.compchemeng.2011.01.031
Salman, H., Yang, G., Zhang, H., Hsieh, C., Zhang, P. (2019). A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networks. Advances in Neural Information Processing Systems 32, 9835–9846.
Schichl, H., Neumaier, A. (2005). Interval Analysis on Directed Acyclic Graphs for Global Optimization. Journal of Global Optimization 33(4), 541–562. doi:10.1007/s10898-005-0937-x
Schulze Darup, M., Mönnigmann, M. (2016). Improved Automatic Computation of Hessian Matrix Spectral Bounds. SIAM Journal on Scientific Computing 38(4), A2068–A2090. doi:10.1137/15m1025773
Schulze Darup, M., Kastsian, M., Mross, S., Mönnigmann, M. (2013). Efficient computation of spectral bounds for Hessian matrices on hyperrectangles for global optimization. Journal of Global Optimization 58(4), 631–652. doi:10.1007/s10898-013-0099-1
Schweidtmann, A. M., Mitsos, A. (2018). Deterministic Global Optimization with Artificial Neural Networks Embedded. Journal of Optimization Theory and Applications 180(3), 925–948. doi:10.1007/s10957-018-1396-0
Schweidtmann, A. M., Bongartz, D., Grothe, D., Kerkenhoff, T., Lin, X., Najman, J., Mitsos, A. (2021). Deterministic global optimization with Gaussian processes embedded. Mathematical Programming Computation 13(3), 553–581. doi:10.1007/s12532-021-00204-y
Scott, J. K., Stuber, M. D., Barton, P. I. (2011). Generalized McCormick relaxations. Journal of Global Optimization 51(4), 569–606. doi:10.1007/s10898-011-9664-7
Shi, Z., Jin, Q., Kolter, Z., Jana, S., Hsieh, C., Zhang, H. (2025). Neural Network Verification with Branch-and-Bound for General Nonlinearities. Tools and Algorithms for the Construction and Analysis of Systems, 315–335. doi:10.1007/978-3-031-90643-5_17
Singh, G., Gehr, T., Püschel, M., Vechev, M. (2019). An abstract domain for certifying neural networks. Proceedings of the ACM on Programming Languages 3(POPL), 1–30. doi:10.1145/3290354
Skjäl, A., Westerlund, T., Misener, R., Floudas, C. A. (2012). A Generalization of the Classical αBB Convex Underestimation via Diagonal and Nondiagonal Quadratic Terms. Journal of Optimization Theory and Applications 154(2), 462–490. doi:10.1007/s10957-012-0033-6
Smith, E., Pantelides, C. (1999). A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Computers & Chemical Engineering 23(4-5), 457–478. doi:10.1016/s0098-1354(98)00286-5
Stanforth, R., Dvijotham, K., Gowal, S., et al. (2020). jax_verify: Neural network verification in JAX. Software, https://github.com/google-deepmind/jax_verify.
Streeter, M., Dillon, J. V. (2022). Automatically Bounding the Taylor Remainder Series: Tighter Bounds and New Applications. https://arxiv.org/abs/2212.11429
Streeter, M., Dillon, J. V. (2023). Sharp Taylor Polynomial Enclosures in One Dimension. https://arxiv.org/abs/2308.00679
Stuber, M. D., Scott, J. K., Barton, P. I. (2014). Convex and concave relaxations of implicit functions. Optimization Methods and Software 30(3), 424–460. doi:10.1080/10556788.2014.924514
Surjanovic, S., Bingham, D. (2013). Virtual Library of Simulation Experiments: Test Functions and Datasets. https://www.sfu.ca/~ssurjano/.
Tawarmalani, M., Sahinidis, N. V. (2005). A polyhedral branch-and-cut approach to global optimization. Mathematical Programming 103(2), 225–249. doi:10.1007/s10107-005-0581-8
Tjandraatmadja, C., Anderson, R., Huchette, J., Ma, W., Patel, K. K., Vielma, J. P. (2020). The Convex Relaxation Barrier, Revisited: Tightened Single-Neuron Relaxations for Neural Network Verification. Advances in Neural Information Processing Systems 33, 21675–21686.
Tsay, C., Kronqvist, J., Thebelt, A., Misener, R. (2021). Partition-based formulations for mixed-integer optimization of trained ReLU neural networks. Advances in Neural Information Processing Systems 34.
Tsoukalas, A., Mitsos, A. (2014). Multivariate McCormick relaxations. Journal of Global Optimization 59(2-3), 633–662. doi:10.1007/s10898-014-0176-0
Vigerske, S., Gleixner, A. (2017). SCIP: global optimization of mixed-integer nonlinear programs in a branch-and-cut framework. Optimization Methods and Software 33(3), 563–593. doi:10.1080/10556788.2017.1335312
Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., et al. (2020). SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods 17(3), 261–272. doi:10.1038/s41592-019-0686-2
Wang, S., Zhang, H., Xu, K., Lin, X., Jana, S., Hsieh, C., Kolter, J. Z. (2021). Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Neural Network Robustness Verification. Advances in Neural Information Processing Systems 34.
Wechsung, A., Scott, J. K., Watson, H. A. J., Barton, P. I. (2015). Reverse propagation of McCormick relaxations. Journal of Global Optimization 63(1), 1–36. doi:10.1007/s10898-015-0303-6
Wilhelm, M. E., Stuber, M. D. (2020). EAGO.jl: easy advanced global optimization in Julia. Optimization Methods and Software 37(2), 425–450. doi:10.1080/10556788.2020.1786566
Wilhelm, M. E., Wang, C., Stuber, M. D. (2022). Convex and concave envelopes of artificial neural network activation functions for deterministic global optimization. Journal of Global Optimization 85(3), 569–594. doi:10.1007/s10898-022-01228-x
Wright, S. J. (1997). Primal-Dual Interior-Point Methods. Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611971453
Wächter, A., Biegler, L. T. (2005). On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming 106(1), 25–57. doi:10.1007/s10107-004-0559-y
Xu, K., Shi, Z., Zhang, H., Wang, Y., Chang, K., Huang, M., Kailkhura, B., Lin, X., et al. (2020). Automatic Perturbation Analysis for Scalable Certified Robustness and Beyond. Advances in Neural Information Processing Systems 33, 1129–1141.
Xu, K., Zhang, H., Wang, S., Wang, Y., Jana, S., Lin, X., Hsieh, C. (2021). Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers. International Conference on Learning Representations (ICLR).
Zhang, H., Weng, T., Chen, P., Hsieh, C., Daniel, L. (2018). Efficient Neural Network Robustness Certification with General Activation Functions. Advances in Neural Information Processing Systems 31.
Zhang, H., Kerkenhoff, T., Kichler, N., Dahmen, M., Mitsos, A., Naumann, U., Bongartz, D. (2025). Accelerating Deterministic Global Optimization via GPU-parallel Interval Arithmetic. doi:10.48550/arXiv.2507.20769